Institute of Information Theory and Automation

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General theories of logical systems

Filozofická fakulta UK
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Algebraic logic is the branch of mathematical logic that studies logical systems by giving them an algebraic semantics. It mainly capitalizes on the standard Linbenbaum–Tarski proof of completeness of classical logic w.r.t. the two-element Boolean algebra, which can be analogously repeated in other logical systems yielding completeness w.r.t. other kinds of algebras. Abstract algebraic logic (AAL) determines what are the essential elements in these proofs and develops an abstract theory of the possible ways in which logical systems can be related to an algebraic counterpart. The usefulness of these methods is witnessed by the fact that the study of many logics, relevant for mathematics, computer science, linguistics or philosophical purposes, has greatly benefited from the algebraic approach, that allows to understand their properties in terms of equivalent algebraic properties of their semantics.


This course is a self-contained introduction to AAL. We start with a focus on the family of weakly implicative logics, i.e. those with a reasonable implication connective, which offer an important conceptual simplification in a yet widely encompassing framework. In a second step, we move to the more general and abstract theory and demonstrate its universal usefulness.



1. Basic syntactical notions. Introduction of important examples (classical logic, intuitionistic logic, Lukasiewicz logic, Gödel-Dummett logic, BCI, and BCK). Completeness w.r.t. the general matrix semantics. Weakly implicative logics and completeness w.r.t. reduced models. Finitary logics and completeness w.r.t. relatively subdirectly irreducible matrices. Characterizations of completeness properties w.r.t. arbitrary classes of models.


2. Generalized disjunctions and proof by cases properties. Semilinear logics and their characterizations. Examples in the family of substructural logics.


3. Leibniz operator on arbitrary logics. Leibniz hierarchy: protoalgebraic, equivalential and (weakly) algebraizable logics. Regularity and finiteness conditions. Alternative characterizations of the classes in the hierarchy. Bridge theorems connecting algebraic and metalogical properties (e.g. deduction theorems, Craig interpolation, Beth definability).




W.J. Blok and D. Pigozzi. Algebraizable logics. Memoirs of the American Mathematical Society 396, vol. 77, 1989.

P. Cintula, P. Hájek, C. Noguera. Handbook of Mathematical Fuzzy Logic – Volume 1, Studies in Logic, Mathematical Logic and Foundations, vol. 37, College Publications, London, 2011.

P. Cintula, C. Noguera. Slabě implikativní logiky: Úvod do abstraktního studia výrokových logik, Univerzita Karlova v Praze, Filozofická fakulta, Prague, 2015.

P. Cintula, C. Noguera. Logic and Implication: An introduction to the general algebraic study of non-classical logics. To appear in Trends in Logic, Springer.

J. Czelakowski. Protoalgebraic Logics. Trends in Logic, vol 10, Dordercht, Kluwer, 2001.

J.M. Font. Abstract Algebraic Logic: An introductory textbook, College Publications, 2016.


2020-01-06 09:24